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\title{\vspace{-2cm} \textbf{第九次课堂作业}}

\author{邵柯欣 \\学号：3200103310 \\课程名称：数据科学的数学基础}

\date{\today}

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\section{$f:\mathbb{R}^n \to \mathbb{R}$是可微凸函数,证明$f(x^*) = min_{x \in \Omega}f(x) \iff \bigtriangledown f^T(x^*)(y-x^*) \ge 0,\forall y \in \Omega$}
Proof:\\
$\because \quad f(x)$是可微凸函数\\
$\therefore \quad $从定义域任意一点做切线，函数图像在切线的上方，即
$$\forall x,y \in \Omega, f(y) \ge f(x) + \bigtriangledown f(x)^T(y - x).$$
$\because \quad f(x^*) = min_{x \in \Omega}f(x)$\\
$$\forall y \in \Omega, \bigtriangledown f(x^*)^T(y - x^*) \ge 0.$$
\section{证明$f(x,y) = x^4+y^4$是严格凸函数}
Proof:\\
$\because \quad X* = (0,0), f(X*) = min_{X \in \mathbb{R}^2}f(X)$并且
$$\forall Y \in \mathbb{R}^2, \bigtriangledown f(X^*)^T(Y - X^*) = 0.$$
$\therefore \quad f(X)$是可微凸函数\\
$$\forall X,Y \in \mathbb{R^2}, f(Y) = f(X) + \bigtriangledown f(X)^T(Y - X) \iff X = Y$$
$\therefore \quad \forall X \ne Y \in \mathbb{R}^2, f(Y) > f(X) + \bigtriangledown f(X)^T(Y - X)$
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